commit 0d50c5f147a78edd2446684c64e642a9550f44ad
parent ec2a57ac212ffa1624d97d50b1531290ee9a74ed
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Tue, 3 Dec 2019 13:20:10 +0100
Fix cite syntax according to template
Diffstat:
1 file changed, 23 insertions(+), 23 deletions(-)
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -19,7 +19,7 @@
% complete documentation is here: http://trackchanges.sourceforge.net/
%%%%%%%
-\usepackage{amsmath}
+%\usepackage{amsmath}
\draftfalse
@@ -58,18 +58,18 @@ TODO
\section{Introduction}
-Fast glacier and ice-sheet flow often ocurrs over weak sedimentary deposits, where basal slip accounts for nearly all movement (e.g., \citeA{Cuffey2010}).
-Basal sediments, called subglacial till, are diamictons commonly consisting of reworked sediments and erosional products (e.g., \citeA{Evans2006}).
-Meltwater fully saturates the pore space, and variations in subglacial water pressure are common and can be caused by internal variability (e.g., \citeA{Kavanaugh2009}) or external water input (e.g., \citeA{Andrews2014, Christoffersen2018}).
-In-situ field observations demonstrate that deformation of this layer can contribute significantly to the glacier movement (e.g., \citeA{Boulton1979, Humphrey1993, Truffer2000}).
+Fast glacier and ice-sheet flow often ocurrs over weak sedimentary deposits, where basal slip accounts for nearly all movement \cite <e.g.,>[]{Cuffey2010}.
+Basal sediments, called subglacial till, are diamictons commonly consisting of reworked sediments and erosional products \cite <e.g.,>[] {Evans2006}.
+Meltwater fully saturates the pore space, and variations in subglacial water pressure are common and can be caused by internal variability \cite<e.g.,>[] {Kavanaugh2009} or external water input \cite<e.g.,>[] {Andrews2014, Christoffersen2018}.
+In-situ field observations demonstrate that deformation of this layer can contribute significantly to the glacier movement \cite<e.g.,>[] {Boulton1979, Humphrey1993, Truffer2000}.
\citeA{Boulton1987} argued that a viscous rheological model with mild stress non-linearity appropriately describes subglacial till deformation.
A viscous rheology implies that the stress required to deform the till is strongly dependent on how fast it is deformed.
However, \citeA{Kamb1991}, \citeA{Iverson1998}, and \citeA{Tulaczyk2000} demonstrated from laboratory shear tests that rate-independent Mohr-Coulomb plasticity, as common for sedimentary materials, is a far better rheological description for subglacial till.
Mohr-Coulomb plastic materials have a yield strength that linearly scales with effective stress and is insensitive to strain rate.
\citeA{Iverson2010} reviewed possible viscous contributions during till-water deformation, but deemed them to be of minor importance.
In spite of a limited observational basis, viscous rheologies continued to be applied as they allow for mathematical modeling of till advection.
-Tills with viscous rheology were used to explain coupled ice-bed processes including subglacial sediment transport (e.g., \citeA{Jenson1995}), landform formation (e.g., \citeA{Hindmarsh1999, Fowler2000}), localization of water drainage (e.g., \citeA{Walder1994, Ng2000b}), and ice-sheet behavior in a changing climate (e.g., \citeA{Pollard2009}).
-Meanwhile, the Mohr-Coulomb plastic model continued to gain further empirical support from laboratory testing (e.g., \citeA{Rathbun2008, Iverson2015}), as well as field observations on mountain glaciers (e.g., \citeA{Hooke1997, Truffer2006, Iverson2007}), mountain glaciers (e.g., \citeA{Kavanaugh2006}), and ice sheets (e.g., \citeA{Tulaczyk2006, Gillet-Chaulet2016, Minchew2016}.
+Tills with viscous rheology were used to explain coupled ice-bed processes including subglacial sediment transport \cite<e.g.,>[] {Jenson1995}, landform formation \cite<e.g.,>[] {Hindmarsh1999, Fowler2000}, localization of water drainage \cite<e.g.,>[] {Walder1994, Ng2000b}, and ice-sheet behavior in a changing climate \cite<e.g.,>[] {Pollard2009}.
+Meanwhile, the Mohr-Coulomb plastic model continued to gain further empirical support from laboratory testing \cite<e.g.,>[] {Rathbun2008, Iverson2015}, as well as field observations on mountain glaciers \cite<e.g.,>[] {Hooke1997, Truffer2006, Iverson2007}, mountain glaciers \cite<e.g.,>[] {Kavanaugh2006}, and ice sheets \cite<e.g.,>[] {Tulaczyk2006, Gillet-Chaulet2016, Minchew2016}.
Inconveniently, the plastic rheology caused a deadlock for the typical continuum modeling of ice and till, as the Mohr-Coulomb constitutive model offers no direct relation between stress and strain rate.
The deadlock was partially resolved when \citeA{Schoof2006} and \citeA{Bueler2009} showed that Mohr-Coulomb friction can be included in ice-sheet models through mathematical reguralization.
While the methods describe the mechanical effect of the bed on the flowing ice, they offer no treatment of sediment erosion, transport, and deposition as strain in the sedimentary bed is not included.
@@ -80,7 +80,7 @@ Unfortunately, intense computational requirements associated with the grain-scal
Instead, simulation of landform to ice-sheet scale requires continuum models.
In this study we build on continuum-modeling advances in granular mechanics and produce a model appropriate for water-saturated sediment deformation in the subglacial environment.
We rely on the original model by \citeA{Henann2013} that was developed for critical state deformation of dry and cohesionless granular materials.
-To correctly account for subglacial tills that are water saturated and often contain a certain amount of cohesion that generally increases with clay content (e.g., \citeA{Iverson1997}), we extend the \citeA{Henann2013} model by including the notion of pore-pressure controlled effective stress, pore-pressure diffusion and add strength contributions from cohesion.
+To correctly account for subglacial tills that are water saturated and often contain a certain amount of cohesion that generally increases with clay content \cite<e.g.,>[] {Iverson1997}, we extend the \citeA{Henann2013} model by including the notion of pore-pressure controlled effective stress, pore-pressure diffusion and add strength contributions from cohesion.
The resultant model contributes the methodological basis required for understanding the coupled dynamics of ice flow over deformable beds.
Different from previous continuum models for till, our model remains true to rheological properties observed in laboratory and field settings.
@@ -88,7 +88,7 @@ In the following, we present the \citeA{Henann2013} model and our modifications
We discuss its applicability and technical limitations before comparing the simulated sediment behavior to published results from laboratory experiments on tills.
The model produces rich dynamics that is consistent and could explain previously poorly-understood field observations.
In particular, the model demonstrates its ability to produce deformation deep away from the ice-bed interface, as occasionally observed in field settings \cite{Truffer2000, Kjaer2006}.
-The model produces stick-slip dynamics under variable water pressures, as observed in mountain glaciers (e.g., \citeA{Fischer1997}) and ice streams (e.g., \citeA{Bindschadler2003}).
+The model produces stick-slip dynamics under variable water pressures, as observed in mountain glaciers \cite<e.g.,>[] {Fischer1997} and ice streams \cite<e.g.,>[] {Bindschadler2003}.
Remnants of pressure deviations within the glacier bed cause hysteresis in stress and strain.
The model source code is constructed with minimal external dependencies, is freely available, and is straight-forward to couple to models of ice-sheet dynamics and glacier hydrology.
@@ -101,15 +101,15 @@ Soils, tills, and other sediments are granular materials, consisting of discrete
This finding evolved into an empirical continuum rheology in \citeA{daCruz2005} and \citeA{Jop2006}, where stress and porosity depend on inertia in a non-linear manner.
However, these continuum models are \emph{local}, meaning that the spatially local state determines the local strain-rate response alone.
Granular deformation contains numerous non-local effects, where flow rates in neighboring areas influence the tendency of a sediment parcel to deform.
-Granular shear zones are an example of the non-locality as they have a minimum width dependent on grain characteristics (e.g., \citeA{Nedderman1980, Forterre2008, Kamrin2018}).
+Granular shear zones are an example of the non-locality as they have a minimum width dependent on grain characteristics \cite<e.g.,>[] {Nedderman1980, Forterre2008, Kamrin2018}.
\subsection{Non-local granular fluidity (NGF) model}%
\label{sub:granular_flow}
\citeA{Henann2013} presented the non-local granular fluidity (NGF) model where a \emph{fluidity} field variable accounts for the non-local effects on deformation.
The model builds on the previous continuum rheology for granular materials by \citeA{daCruz2005} and \citeA{Jop2006}, and with non-local effects accurately describes strain distribution in a variety of experimental settings.
In the \citeA{Henann2013} model, all material is assumed to have a uniform porosity and be in the critical state.
-Fluidity acts as a state variable, describing the phase transition between non-deforming (jammed) and actively deforming (flowing) parts of the sediment (e.g., \citeA{Zhang2017}).
-The modeled sediment deforms as a highly nonlinear Bingham material with yield beyond the Mohr-Coulomb failure limit (e.g., \citeA{Henann2013, Henann2016}), but unlike classical plastic models it includes a closed form relation that predicts the stress-strain rate relation beyond yield.
+Fluidity acts as a state variable, describing the phase transition between non-deforming (jammed) and actively deforming (flowing) parts of the sediment \cite<e.g.,>[] {Zhang2017}.
+The modeled sediment deforms as a highly nonlinear Bingham material with yield beyond the Mohr-Coulomb failure limit \cite<e.g.,>[] {Henann2013, Henann2016}, but unlike classical plastic models it includes a closed form relation that predicts the stress-strain rate relation beyond yield.
For the purposes in this paper, we assume that plastic shear strain ($\dot{\gamma}^\mathrm{p}$) contributes all of the resultant deformation ($\dot{\gamma}$):
\begin{linenomath*}
\begin{equation}
@@ -139,10 +139,10 @@ The local contribution to fluidity is defined as:
\begin{linenomath*}
\begin{equation}
g_\mathrm{local}(\mu, \sigma_\mathrm{n}') =
- \begin{cases}
- \sqrt{d^2 \sigma_\mathrm{n}' / \rho_\mathrm{s}} ((\mu - C/\sigma_\mathrm{n}') - \mu_\mathrm{s})/(b\mu) &\mathrm{if } \mu - C/\sigma_\mathrm{n}' > \mu_\mathrm{s} \mathrm{, and}\\
- 0 & \mathrm{if } \mu - C/\sigma_\mathrm{n}' \leq \mu_\mathrm{s}.
- \end{cases}
+ %\begin{cases}
+ %\sqrt{d^2 \sigma_\mathrm{n}' / \rho_\mathrm{s}} ((\mu - C/\sigma_\mathrm{n}') - \mu_\mathrm{s})/(b\mu) &\mathrm{if } \mu - C/\sigma_\mathrm{n}' > \mu_\mathrm{s} \mathrm{, and}\\
+ %0 & \mathrm{if } \mu - C/\sigma_\mathrm{n}' \leq \mu_\mathrm{s}.
+ %\end{cases}
\label{eq:g_local}
\end{equation}
\end{linenomath*}
@@ -153,7 +153,7 @@ This characteristic also strengthens the material if the shear zone size is rest
\subsection{Fluid-pressure evolution}%
\label{sub:fluid_pressure_evolution}
-We prescribe the transient evolution of pore-fluid pressure ($p_\mathrm{f}$) by Darcian pressure diffusion (e.g., \citeA{Goren2010, Goren2011, Damsgaard2017b}):
+We prescribe the transient evolution of pore-fluid pressure ($p_\mathrm{f}$) by Darcian pressure diffusion \cite<e.g.,>[] {Goren2010, Goren2011, Damsgaard2017b}:
\begin{linenomath*}
\begin{equation}
\frac{\partial p_\mathrm{f}}{\partial t} = \frac{1}{\phi\eta_\mathrm{f}\beta_\mathrm{f}} \nabla \cdot (k \nabla p_\mathrm{f}),
@@ -169,7 +169,7 @@ The local fluid pressure is used to determine the effective normal stress used i
\label{sub:limitations_of_the_continuum_model}
The presented model considers the material to be in the critical (steady) state throughout the domain.
Consequently, porosity is prescribed as a constant and material-specific parameter.
-For that reason the model is not able to simulate uniaxial compaction or shear-induced volume changes (e.g., \citeA{Iverson2000, Iverson2010-2, Damsgaard2015}) or compaction and dilation (e.g., \citeA{Dewhurst1996}).
+For that reason the model is not able to simulate uniaxial compaction or shear-induced volume changes \cite<e.g.,>[] {Iverson2000, Iverson2010b, Damsgaard2015} or compaction and dilation \cite<e.g.,>[] {Dewhurst1996}.
We currently have a transient granular continuum model with state-dependent porosity under development.
However, \citeA{Iverson2010} argued that the majority of actively deforming subglacial sediment may be in the critical state.
For that reason, and due to the emerging insights from the current simple model, we see this contribution as a valuable first step that allows us to isolate the dynamic that emerges in the critical state.
@@ -178,7 +178,7 @@ In the NGF model, the representative grain size $d$ scales the non-locality and
The relation between the physical grain size distribution and the effective grain size $d$ that controls the fluidity has so far not been sufficiently explored.
For the case of till layers, that are characterized by a fractal grain size distribution \cite{Hooke1995}, the relation to the fluidity is even more obscure.
The issue is left for future research and here we assume that $d$ corresponds to the volumetrically dominant grain size, if one exists.
-Specifically designed laboratory experiments with various tills should inform the treatment of length scale, outside of ploughing effects by large clasts protruding from the basal ice (e.g., \citeA{Tulaczyk1999}).
+Specifically designed laboratory experiments with various tills should inform the treatment of length scale, outside of ploughing effects by large clasts protruding from the basal ice \cite<e.g.,>[] {Tulaczyk1999}.
\subsection{Simulation setup}
Parameter values and their references are listed in Table~S1.
@@ -186,7 +186,7 @@ For the first experiment with variable water pressure, we apply a water-pressure
Many simulations are performed under both stress- and rate-controlled shear, which both idealize the driving glacier physics.
Real glacier settings fall somewhere in between, depending on how important basal friction is to the overall stress balance.
Stress-controlled conditions approximate a setting where ice flow directly responds to changes in subglacial strain rates.
-Whillans Ice Plain, West Antarctica is an example of this setting, where a low surface slope and low driving stress results in stick-slip movement (e.g., \citeA{Bindschadler2003}).
+Whillans Ice Plain, West Antarctica is an example of this setting, where a low surface slope and low driving stress results in stick-slip movement \cite<e.g.,>[] {Bindschadler2003}.
A rate-controlled setup is the opposite end member, where changes in bed friction do not influence ice flow velocity.
\begin{figure*}[htbp]
@@ -306,13 +306,13 @@ Regardless of perturbation shape, the maximum deformation depth increases with i
\section{Discussion}%
\label{sec:discussion}
In this study it is assumed that there is a strong coupling between ice and bed.
-However, overpressurization and slip at the ice-bed interface may cause episodic decoupling at the interface and reduce bed deformation, as observed under Whillans Ice Stream, West Antarctica (e.g., \citeA{Engelhardt1998}), and in deposits from Pleistocene glaciations (e.g., \citeA{Piotrowski2001}).
+However, overpressurization and slip at the ice-bed interface may cause episodic decoupling at the interface and reduce bed deformation, as observed under Whillans Ice Stream, West Antarctica \cite<e.g.,>[] {Engelhardt1998}, and in deposits from Pleistocene glaciations \cite<e.g.,>[] {Piotrowski2001}.
We see the presented framework as a significant improvement of treating sediment advection in ice-flow models, but acknowledge that a complete understanding of the sediment mass budget requires improved models of ice-bed interface physics.
The stress-dependent sediment advection without variations in the pore pressure observed in Fig.~\ref{fig:strain_distribution} is relevant for instability theories of subglacial landform development \cite{Hindmarsh1999, Fowler2000, Schoof2007, Fowler2018}.
From geometrical considerations, it is likely that bed-normal stresses on the stoss side of subglacial topography are higher than on the lee side.
With all other physical conditions being equal, our results indicate that shear-driven sediment advection would be larger on the stoss side of bed perturbations than behind them.
-Topography of non-planar ice-bed interfaces (proto-drumlins, ribbed moraines, etc.) may be transported and modulated through this variable transport capacity, unless stress differences are overprinted by spatial variations in water pressure (e.g., \citeA{Sergienko2013, McCracken2016, Iverson2017b, Hermanowski2019b}).
+Topography of non-planar ice-bed interfaces (proto-drumlins, ribbed moraines, etc.) may be transported and modulated through this variable transport capacity, unless stress differences are overprinted by spatial variations in water pressure \cite<e.g.,>[] {Sergienko2013, McCracken2016, Iverson2017b, Hermanowski2019b}.
At depth, the water pressure variations display exponential decay in amplitude and increasing lag.
As long as fluid and diffusion properties are constant and the layer is sufficiently thick, an analytical solution to skin depth $d_\mathrm{s}$ [m] in our system follows the form (after Eq.~4.90 in \citeA{Turcotte2002}),
@@ -336,7 +336,7 @@ It is worth noting that to induce deep deformation the water pressure deviations
This means that minima in effective normal stress are increasingly difficult to create at larger depths through pure diffusion from the ice-bed interface.
Due to higher hydraulic permeability, coarse tills are more susceptible to deep deformation, but deep strain requires longer-lasting perturbations in water pressure (Fig.~\ref{fig:skin_depth}).
Contrarily, fine-grained tills are unlikely to cause deep deformation.
-Lateral water input at depth may be a viable alternate mechanism for creating occasional episodes of deep slip, in particular when horizontal bedding decreases vertical permeability (e.g., \citeA{Kjaer2006}).
+Lateral water input at depth may be a viable alternate mechanism for creating occasional episodes of deep slip, in particular when horizontal bedding decreases vertical permeability \cite<e.g.,>[] {Kjaer2006}.
%TODO: LAKE DRAINAGE
\section{Conclusion}%
